In an important recent working paper, M. Fabinger and E. G. Weyl characterize
tractable monopoly problems. \(^{18}\) A "tractable" problem satisfies three
conditions. First, it must be possible to move back and forth between explicit
expressions for inverse and direct demand (invertibility). Second, inverse
demand-which can also be interpreted as average revenuemust have the same
functional form as marginal revenue, and average cost must have the same
functional form as marginal cost (form preservation). Third, the monopolist's
first-order condition must be a linear equation (linearity), if not
immediately after differentiation, then at least after suitable substitution.
The authors show that the broadest possible class of tractable problems has
the following functional form for inverse demand and average cost:
\\[
\begin{aligned}
P(Q) &=a_{0}+a_{1} Q^{-s} \\
A C(Q) &=c_{0}+c_{1} Q^{-s}
\end{aligned}
\\]
where \(a_{0}, a_{1}, c_{0}, c_{1},\) and \(s\) are non-negative constants.
a. Solve for the monopoly equilibrium quantity and price given these
functional forms. What substitution \(x=f(Q)\) do you need to make the first-
order condition linear in \(x ?\)
b. Derive the solution in the special case with constant average and marginal
cost. c. If one is willing to relax tractability a bit to allow the monopoly's
first-order condition to be a quadratic equation (at least after suitable
substitution), the authors show that the broadest class of tractable problems
then involves the following functional forms:
\\[
\begin{aligned}
P(Q) &=a_{0}+a_{1} Q^{-s}+a_{2} Q^{s} \\
A C(Q) &=c_{0}+c_{1} Q^{-s}+c_{2} Q^{s}
\end{aligned}
\\]
Solve for the monopoly equilibrium quantity and price. What substitution
\(x=f(Q)\) is needed to make the first-order condition quadratic in \(x ?\)
d. While slightly complicated, the functional forms in part (c) have the
advantage of being flexible enough to allow for U-shaped average cost curves
such as drawn in Figure 14.2 in addition to constant, increasing, and
decreasing. Demonstrate this by graphing this average cost curve for well-
chosen values of \(c_{0}, c_{1}, c_{2}\) to illustrate the various cases. The
flexible functional forms in part (c) also allow for realistic demand shapes,
for example, one that closely fits the U.S. income distribution (which
implicitly takes income to proxy for consumers' willingness to pay). These
realistic demand shapes can be used in calibrations to address important
policy questions. For example, the text mentioned that, in theory, the welfare
effects of monopoly price discrimination can
go either way, either being higher or lower than under uniform pricing.
Calibrations involving the demand curves from part
(c) invariably show that welfare is higher under price discrimination.
